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This work deals with a system of interacting reinforced stochastic processes,where each process $X^j=(X_{n,j})_n$ is located at a vertex $j$ of a finiteweighted direct graph, and it can be interpreted as the sequence of "actions"adopted by an agent $j$ of the network. The interaction among the dynamics ofthese processes depends on the weighted adjacency matrix $W$ associated to theunderlying graph: indeed, the probability that an agent $j$ chooses a certainaction depends on its personal "inclination" $Z_{n,j}$ and on the inclinations$Z_{n,h}$, with $h\neq j$, of the other agents according to the entries of $W$.The best known example of reinforced stochastic process is the Polya urn.  The present paper characterizes the asymptotic behavior of the weightedempirical means $N_{n,j}=\sum_{k=1}^n q_{n,k} X_{k,j}$, proving their almostsure synchronization and some central limit theorems in the sense of stableconvergence. By means of a more sophisticated decomposition of the consideredprocesses adopted here, these findings complete and improve some asymptoticresults for the personal inclinations $Z^j=(Z_{n,j})_n$ and for the empiricalmeans $\overline{X}^j=(\sum_{k=1}^n X_{k,j}/n)_n$ given in recent papers (e.g.[arXiv:1705.02126, Bernoulli, Forth.]; [arXiv:1607.08514, Ann. Appl. Probab.,27(6):3787-3844, 2017]; [arXiv:1602.06217, Stochastic Process. Appl.,129(1):70-101, 2019]). Our work is motivated by the aim to understand how thedifferent rates of convergence of the involved stochastic processes combineand, from an applicative point of view, by the construction of confidenceintervals for the common limit inclination of the agents and of a teststatistics to make inference on the matrix $W$, based on the weighted empiricalmeans. In particular, we answer a research question posed in [arXiv:1705.02126,Bernoulli, Forth.]

Interacting reinforced stochastic processes: statistical inference based on the weighted empirical means